$\beta \beta$ Decay Nuclear Matrix Elements

As a part of recent studies into the nature of the neutrino, searches for the neutrinoless ($0\nu$) mode of double beta ($\beta\beta$) decay are among the most exciting since this process violates lepton-number conservation and would establish the neutrino as a Majorana particle [1]. Several collaborations are currently performing large scale searches using a series of different isotopes. Among these experiments are KamLAND-ZEN [2] and EXO-200 [3]
($^{136}$Xe), GERDA [4] and MAJORANA [5] ($^{76}$Ge), NEMO-III [6] ($^{100}$Mo, $^{116}$Cd, $^{130}$Te), and SNO+ [7] ($^{130}$Te). If in these searches the $0\nu$ decay mode is observed (under the assumption of light neutrino exchange), the effective Majorana mass of the neutrino, $m_{\beta\beta}$, can be deduced from $0\nu\beta\beta$ measurements, \begin{eqnarray} \label{0nubb} (T^{0\nu}_{1/2})^{-1}=G^{0\nu}(Q,Z)|M^{0\nu}|^2\langle m_{\beta\beta}\rangle^2, \end{eqnarray} where $T^{0\nu}_{1/2}$ is the observed half-life of the $0\nu\beta\beta$ decay and $G^{0\nu}(Q,Z)$ is the phase-space factor. The term $M^{0\nu}$ is the nuclear matrix element (NME) connecting the initial and final $0^+$ states, which results entirely from theoretical calculations [8]. Currently, NMEs calculated within different theoretical frameworks (and sometimes even within the same framework by different groups) vary significantly, as shown in Fig. 1 (top). This spread in the calculated NMEs has introduced large uncertainties to the extracted limits on the effective Majorana neutrino mass from $0\nu\beta\beta$ decay direct searches. Further, since no direct benchmarking of the calculations have been possible to date (since the $0\nu$ mode has not been observed), there is a concern that even this spread may not encapsulate the correct values. As a result of the current discrepancy in the NME calculations, no reliable limit on the effective Majorana neutrino mass can be extracted from the direct $0\nu\beta\beta$ decay searches (Fig. 1 (bottom)). Due to the significance of determining the absolute mass scale of the neutrinos, this is a major concern. Typically, the NME calculations are benchmarked to $2\nu\beta\beta$ data [9] (a process allowed by the Standard Model) where the decay path proceeds through $1^+$ states in the odd-odd intermediate nucleus [10]. These GT transitions ($0^+\rightarrow1^+$) have been probed in the past using high resolution charge-exchange reactions on the relevant $\beta\beta$ decay nuclei (for example $^{128}$Te and $^{130}$Te [11]). However, these approaches can be limited by the model-dependent nature of the transfer reaction mechanism, and the subsequent analysis required to extract the GT strength [12].

allowedallowed
Figure 1: (top) A comparison of the calculated $M'^{0\nu}$ values for five $\beta\beta$ decay candidates from the different theoretical methods ([14]). (bottom) Current limits of the EXO-200, KamLAND Zen, and GERDA $0\nu\beta\beta$ decay searches [3]. The half-life limits of $^{76}$Ge and $^{136}$Xe are compared to each other through the effective Majorana neutrino mass which is drawn as half-life limits of $^{76}$Ge and $^{136}$Xe are compared to each other through the effective Majorana neutrino mass which is drawn as diagonal lines. The different lines illustrate the effect that the spread in the NMEs have on the extracted limits of the Majorana neutrino mass based on the current $^{76}$Ge and $^{136}$Xe $0\nu\beta\beta$ decay $T_{1/2}$ limits.


An alternative, less model-dependent approach towards providing detailed information to $2\nu\beta\beta$ NME calculations is by measuring the weak GT decay branches of the radioactive intermediate nuclei themselves. Additionally, in special cases, where a so-called single-state dominance (SSD) is present, the transition via the lowest $1^+$ state accounts for the whole NME [13]. Therefore, measurements of the electron-capture (EC) branching ratios (BR) of the intermediate nuclei in the $2\nu\beta\beta$ process are directly able to capture the nuclear-physics information required. Typically the EC transitions are several orders of magnitude weaker than the dominant $\beta$ decays from the same parent nucleus, making them difficult to detect. To circumvent this challenge, we have jointly developed a low-background, high-sensitivity decay spectroscopy tool in collaboration with the TITAN group at TRIUMF to measure characteristic X-rays from weak EC decays [10]. We maintain a strong, active collaboration with the TITAN ion-trap system at TRIUMF, and contribute to its many experiments.

Further nuclear structure studies on these $\beta\beta$ decay systems are critical to benchmark the theoretical calculations used to extract the NMEs. We are currently working on using our experiences with detailed studies from transfer reactions on the superallowed nuclei to provide a more complete picture on the structure of these nuclei.
Citations
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